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November  2013, 7(4): 425-440. doi: 10.3934/amc.2013.7.425

On the dual of (non)-weakly regular bent functions and self-dual bent functions

 1 Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany, Germany 2 MDBF, Sabanci University, Orhanlı, Tuzla 34956, İstanbul, Turkey

Received  July 2012 Revised  March 2013 Published  October 2013

For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of non-weakly regular bent functions and show conditions under which their dual is bent as well. This leads to the definition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a different algebraic degree.
Citation: Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425
References:
 [1] C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28.  Google Scholar [2] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.  Google Scholar [3] C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399. doi: 10.1504/IJICOT.2010.032864.  Google Scholar [4] C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681.  Google Scholar [5] A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429. doi: 10.1016/j.jcta.2011.10.002.  Google Scholar [6] A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242. doi: 10.1007/s10623-012-9686-2.  Google Scholar [7] Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266. doi: 10.1007/s10801-010-0270-4.  Google Scholar [8] Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103.  Google Scholar [9] K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090. doi: 10.1016/j.ffa.2008.07.002.  Google Scholar [10] F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203. doi: 10.1007/978-3-642-15874-2_16.  Google Scholar [11] T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854.  Google Scholar [12] T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130.  Google Scholar [13] T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291. doi: 10.1007/s12095-011-0048-0.  Google Scholar [14] X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198. doi: 10.1007/s10623-011-9544-7.  Google Scholar [15] K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013. doi: 10.1109/TIT.2010.2050806.  Google Scholar [16] N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.  Google Scholar [17] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997.  Google Scholar [18] Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003.  Google Scholar [19] Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159. Google Scholar

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References:
 [1] C. Carlet, On the secondary constructions of resilient and bent functions, in Coding, Cryptography and Combinatorics, Birkhäuser Basel, 2004, 3-28.  Google Scholar [2] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.  Google Scholar [3] C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. Coding Theory, 1 (2010), 384-399. doi: 10.1504/IJICOT.2010.032864.  Google Scholar [4] C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681.  Google Scholar [5] A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory Ser. A, 119 (2012), 420-429. doi: 10.1016/j.jcta.2011.10.002.  Google Scholar [6] A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013), 231-242. doi: 10.1007/s10623-012-9686-2.  Google Scholar [7] Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions, J. Algebr. Comb., 34 (2011), 251-266. doi: 10.1007/s10801-010-0270-4.  Google Scholar [8] Y. Edel and A. Pott, On the equivalence of nonlinear functions, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Amsterdam, 2009, 87-103.  Google Scholar [9] K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12, Finite Fields Appl., 14 (2008), 1083-1090. doi: 10.1016/j.ffa.2008.07.002.  Google Scholar [10] F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), Springer-Verlag, Berlin, 2010, 196-203. doi: 10.1007/978-3-642-15874-2_16.  Google Scholar [11] T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854.  Google Scholar [12] T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130.  Google Scholar [13] T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun., 3 (2011), 281-291. doi: 10.1007/s12095-011-0048-0.  Google Scholar [14] X. D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63 (2012), 183-198. doi: 10.1007/s10623-011-9544-7.  Google Scholar [15] K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums, IEEE Trans. Inform. Theory, 56 (2010), 4011-4013. doi: 10.1109/TIT.2010.2050806.  Google Scholar [16] N. G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.  Google Scholar [17] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Cambridge Univ. Press, Cambridge, 1997.  Google Scholar [18] Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Comb. Theory Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003.  Google Scholar [19] Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular, in Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, 156-159. Google Scholar
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